All of the 3rd grade teachers and students from Covington went on a field trip to an art museum. Tickets were $$6.50$ each for teachers and $$3.00$ each for students, and the group paid $$56.00$ in total. A few weeks later, the same group visited a natural history museum where the tickets cost $$26.00$ each for teachers and $$9.50$ each for students, and the group paid $$199.00$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${6.5x+3y = 56}$ ${26x+9.5y = 199}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-26x-12y = -224}$ ${26x+9.5y = 199}$ Add the top and bottom equations together. $ -2.5y = -25 $ $ y = \dfrac{-25}{-2.5}$ ${y = 10}$ Now that you know ${y = 10}$ , plug it back into $ {6.5x+3y = 56}$ to find $x$ ${6.5x + 3}{(10)}{= 56}$ $6.5x+30 = 56$ $6.5x = 26$ $x = \dfrac{26}{6.5}$ ${x = 4}$ You can also plug ${y = 10}$ into $ {26x+9.5y = 199}$ and get the same answer for $x$ ${26x + 9.5}{(10)}{= 199}$ ${x = 4}$ There were $4$ teachers and $10$ students on the field trips.